AP Statistics z-Scores and the Normal Distribution Worksheet Solutions (PDF)
This ap statistics normal distribution answers for AP Statistics covers z-scores and the normal distribution (Common Core AP STAT 1.10, AP STAT 1.11, HSS-ID.A.4), with a free PDF to print.
AP Statistics · Common Core State Standards: AP STAT 1.10, AP STAT 1.11, HSS-ID.A.4
Ap statistics normal distribution answers: exercises 1 to 10
Exercise 1 – Compute a z-score
\(z=\frac{69-64}{2.5}=2.00\).
z = 2.00; she is 2 standard deviations above the mean.
Exercise 2 – Value from a z-score
\(x=\mu+z\sigma=78+(-1.5)(6)=69\).
69.
Exercise 3 – Compare two scores
Jamal: \(z=\frac{84-75}{6}=1.5\). Sofia: \(z=\frac{640-600}{50}=0.8\).
Jamal did relatively better (z = 1.5 compared with 0.8).
Exercise 4 – Empirical rule
420 and 580 are 2 standard deviations from 500, so about 95%.
460 and 540 are 1 standard deviation away, so about 68%.
About 95%; about 68%.
Exercise 5 – Empirical rule counts
15 is 2 standard deviations above the mean. About 95% are within 2, so 5% are outside, and half of those, 2.5%, are above.
\(0.025\times 400=10\).
About 10 batteries.
Exercise 6 – Area to the left
\(z=\frac{280-250}{20}=1.5\). From the table or calculator, \(P(Z < 1.5)=0.9332\).
0.9332, about 93.3%.
Exercise 7 – Area to the right
\(z=\frac{215-250}{20}=-1.75\). \(P(Z > -1.75)=1-0.0401=0.9599\).
0.9599.
Exercise 8 – Area between
The boundaries have z = -1 and z = 1. \(0.8413-0.1587=0.6827\).
0.6827, matching the 68% of the empirical rule.
Exercise 9 – Value from a percentile
For area 0.90 to the left, \(z=1.28\). \(x=100+(1.28)(15)=119.2\).
About 119.2.
Exercise 10 – Percentile of a value
\(z=\frac{22-27}{4}=-1.25\). \(P(Z < -1.25)=0.1056\).
About the 10.6th percentile: about 10.6% of runners were faster than 22 minutes. (A lower time is faster, so 10.6% of runners were at or below her time.)
Exercises 11 and up
Exercise 11 – Quartiles of a normal
\(z_{Q_1}=-0.6745,\; z_{Q_3}=0.6745\).
\(Q_1=100-0.6745(15)=89.9\), \(Q_3=110.1\).
Q1 = 89.9, Q3 = 110.1, IQR = 20.2.
Exercise 12 – Change of units
In centimeters: mean 162.56, sd 6.35, height 175.26. \(z=\frac{175.26-162.56}{6.35}=2.00\).
z = 2; z-scores do not change under a change of units.
Exercise 13 – Is it approximately normal?
Within 1 sd: \(\frac{138}{200}=69\%\). Within 2 sd: \(\frac{192}{200}=96\%\).
These are close to 68% and 95%, so the data are approximately normal (a histogram should also be checked).
Exercise 14 – Bottle filling
\(z=\frac{11.8-12}{0.15}=-1.33\). \(P(Z < -1.33)=0.0918\).
\(0.0918\times 5000=459\).
About 9.18%; about 459 bottles.
Exercise 15 – Tall women
\(z=\frac{70-64}{2.5}=2.4\). \(P(Z > 2.4)=1-0.9918=0.0082\).
About 0.82%. It is unusual (z = 2.4 is beyond 2 standard deviations).
Exercise 16 – Commute times
\(z=\frac{55-60}{4}=-1.25\) and \(z=1.25\). \(0.8944-0.1056=0.7887\).
About 78.9%.
Exercise 17 – Top 5 percent
The 95th percentile has \(z=1.645\).
\(x=100+1.645(15)=124.7\).
About 124.7, so an IQ of at least 125 (rounded up).
Exercise 18 – Find the standard deviation
2.5% above means 4.0 is 2 standard deviations above the mean.
\(2\sigma=4.0-3.2=0.8\), so \(\sigma=0.4\).
About 0.4 oz.
Exercise 19 – Two-step
Within 0.5: \(z=\pm 0.625\), \(P=0.4680\).
More than 1 inch above: \(z=1.25\), \(P=0.1056\).
About 46.8% and 10.6%.
Exercise 20 – Scores above a cutoff
\(z=\frac{90-72}{9}=2\). \(P(Z > 2)=1-0.9772=0.0228\).
About 2.3%.
Key facts behind this ap statistics normal distribution answers
- z = (x – mean) / sd measures position in standard deviations.
- Empirical rule: 68% within 1 sd, 95% within 2 sd, 99.7% within 3 sd.
- Area under the curve over an interval is the proportion of values in it; total area is 1.
- Use normalcdf for areas and invNorm for percentiles; always sketch and shade first.
- A percentile is the percent of values at or below a given value.
How to use this ap statistics normal distribution answers
Cover each solution, solve the exercise again, then uncover it and compare every line. If your answer differs, find the first step that changed and redo the exercise from there.
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